www.elsolucionario.net www.elsolucionario.net Feedback Systems An Introduction for Scientists and Engineers ˚ Karl Johan Aström Richard M Murray Version v2.10e (30 August 2011) This is the electronic edition of Feedback Systems and is available from http://www.cds.caltech.edu/∼murray/amwiki Hardcover editions may be purchased from Princeton Univeristy Press, http://press.princeton.edu/titles/8701.html This manuscript is for personal use only and may not be reproduced, in whole or in part, without written consent from the publisher (see http://press.princeton.edu/permissions.html) PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD www.elsolucionario.net Copyright © 2008 by Princeton University Press Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Åström, Karl J (Karl Johan), 1934Feedback systems : an introduction for scientists and engineers / Karl Johan Åström and Richard M Murray p cm Includes bibliographical references and index ISBN-13: 978-0-691-13576-2 (alk paper) ISBN-10: 0-691-13576-2 (alk paper) Feedback control systems I Murray, Richard M., 1963- II Title TJ216.A78 2008 629.8′ 3–dc22 2007061033 British Library Cataloging-in-Publication Data is available This book has been composed in LATEX The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper ∞ press.princeton.edu Printed in the United States of America 10 www.elsolucionario.net ii This version of Feedback Systems is the electronic edition of the text Revision history: • Version 2.10e (30 Aug 2011): electronic edition, with corrections • Version 2.10d (19 Jul 2011): electronic edition, with corrections • Version 2.10c (4 Mar 2010): third printing, with corrections • Version 2.10b (22 Feb 2009): second printing, with corrections • Version 2.10a (2 Dec 2008): electronic edition, with corrections • Version 2.9d (30 Jan 2008): first printing A full list of changes made in each revision is available on the companion web site: http://www.cds.caltech.edu/^murray/FBSwiki www.elsolucionario.net Contents Preface vi Chapter Introduction 1.1 What Is Feedback? 1.2 What Is Control? 1.3 Feedback Examples 1.4 Feedback Properties 1.5 Simple Forms of Feedback 1.6 Further Reading Exercises 1 17 23 25 26 Chapter System Modeling 2.1 Modeling Concepts 2.2 State Space Models 2.3 Modeling Methodology 2.4 Modeling Examples 2.5 Further Reading Exercises 27 27 34 44 51 61 61 Chapter Examples 3.1 Cruise Control 3.2 Bicycle Dynamics 3.3 Operational Amplifier Circuits 3.4 Computing Systems and Networks 3.5 Atomic Force Microscopy 3.6 Drug Administration 3.7 Population Dynamics Exercises 66 66 70 72 76 82 85 90 92 Chapter Dynamic Behavior 4.1 Solving Differential Equations 4.2 Qualitative Analysis 4.3 Stability 4.4 Lyapunov Stability Analysis 4.5 Parametric and Nonlocal Behavior 96 96 99 103 111 121 www.elsolucionario.net iv CONTENTS 4.6 Further Reading Exercises 127 127 Chapter Linear Systems 5.1 Basic Definitions 5.2 The Matrix Exponential 5.3 Input/Output Response 5.4 Linearization 5.5 Further Reading Exercises 132 132 137 146 159 164 165 Chapter State Feedback 6.1 Reachability 6.2 Stabilization by State Feedback 6.3 State Feedback Design 6.4 Integral Action 6.5 Further Reading Exercises 168 168 176 184 196 198 199 Chapter Output Feedback 7.1 Observability 7.2 State Estimation 7.3 Control Using Estimated State 7.4 Kalman Filtering 7.5 A General Controller Structure 7.6 Further Reading Exercises 202 202 207 212 216 220 227 227 Chapter Transfer Functions 8.1 Frequency Domain Modeling 8.2 Derivation of the Transfer Function 8.3 Block Diagrams and Transfer Functions 8.4 The Bode Plot 8.5 Laplace Transforms 8.6 Further Reading Exercises 230 230 232 243 251 260 263 263 Chapter Frequency Domain Analysis 9.1 The Loop Transfer Function 9.2 The Nyquist Criterion 9.3 Stability Margins 9.4 Bode’s Relations and Minimum Phase Systems 9.5 Generalized Notions of Gain and Phase 9.6 Further Reading 269 269 272 280 285 287 292 www.elsolucionario.net v CONTENTS Exercises 293 Chapter 10 PID Control 10.1 Basic Control Functions 10.2 Simple Controllers for Complex Systems 10.3 PID Tuning 10.4 Integrator Windup 10.5 Implementation 10.6 Further Reading Exercises 296 296 301 305 309 311 316 316 Chapter 11 Frequency Domain Design 11.1 Sensitivity Functions 11.2 Feedforward Design 11.3 Performance Specifications 11.4 Feedback Design via Loop Shaping 11.5 Fundamental Limitations 11.6 Design Example 11.7 Further Reading Exercises 319 319 323 326 330 335 344 347 348 Chapter 12 Robust Performance 12.1 Modeling Uncertainty 12.2 Stability in the Presence of Uncertainty 12.3 Performance in the Presence of Uncertainty 12.4 Robust Pole Placement 12.5 Design for Robust Performance 12.6 Further Reading Exercises 351 351 356 362 365 373 378 378 Bibliography 381 Index 391 www.elsolucionario.net Preface This book provides an introduction to the basic principles and tools for the design and analysis of feedback systems It is intended to serve a diverse audience of scientists and engineers who are interested in understanding and utilizing feedback in physical, biological, information and social systems We have attempted to keep the mathematical prerequisites to a minimum while being careful not to sacrifice rigor in the process We have also attempted to make use of examples from a variety of disciplines, illustrating the generality of many of the tools while at the same time showing how they can be applied in specific application domains A major goal of this book is to present a concise and insightful view of the current knowledge in feedback and control systems The field of control started by teaching everything that was known at the time and, as new knowledge was acquired, additional courses were developed to cover new techniques A consequence of this evolution is that introductory courses have remained the same for many years, and it is often necessary to take many individual courses in order to obtain a good perspective on the field In developing this book, we have attempted to condense the current knowledge by emphasizing fundamental concepts We believe that it is important to understand why feedback is useful, to know the language and basic mathematics of control and to grasp the key paradigms that have been developed over the past half century It is also important to be able to solve simple feedback problems using back-of-the-envelope techniques, to recognize fundamental limitations and difficult control problems and to have a feel for available design methods This book was originally developed for use in an experimental course at Caltech involving students from a wide set of backgrounds The course was offered to undergraduates at the junior and senior levels in traditional engineering disciplines, as well as first- and second-year graduate students in engineering and science This latter group included graduate students in biology, computer science and physics Over the course of several years, the text has been classroom tested at Caltech and at Lund University, and the feedback from many students and colleagues has been incorporated to help improve the readability and accessibility of the material Because of its intended audience, this book is organized in a slightly unusual fashion compared to many other books on feedback and control In particular, we introduce a number of concepts in the text that are normally reserved for secondyear courses on control and hence often not available to students who are not control systems majors This has been done at the expense of certain traditional topics, which we felt that the astute student could learn independently and are often www.elsolucionario.net vii PREFACE explored through the exercises Examples of topics that we have included are nonlinear dynamics, Lyapunov stability analysis, the matrix exponential, reachability and observability, and fundamental limits of performance and robustness Topics that we have deemphasized include root locus techniques, lead/lag compensation and detailed rules for generating Bode and Nyquist plots by hand Several features of the book are designed to facilitate its dual function as a basic engineering text and as an introduction for researchers in natural, information and social sciences The bulk of the material is intended to be used regardless of the audience and covers the core principles and tools in the analysis and design of feedback systems Advanced sections, marked by the “dangerous bend” symbol shown here, contain material that requires a slightly more technical background, of the sort that would be expected of senior undergraduates in engineering A few sections are marked by two dangerous bend symbols and are intended for readers with more specialized backgrounds, identified at the beginning of the section To limit the length of the text, several standard results and extensions are given in the exercises, with appropriate hints toward their solutions To further augment the printed material contained here, a companion web site has been developed and is available from the publisher’s web page: http://www.cds.caltech.edu/∼murray/amwiki The web site contains a database of frequently asked questions, supplemental examples and exercises, and lecture material for courses based on this text The material is organized by chapter and includes a summary of the major points in the text as well as links to external resources The web site also contains the source code for many examples in the book, as well as utilities to implement the techniques described in the text Most of the code was originally written using MATLAB M-files but was also tested with LabView MathScript to ensure compatibility with both packages Many files can also be run using other scripting languages such as Octave, SciLab, SysQuake and Xmath The first half of the book focuses almost exclusively on state space control systems We begin in Chapter with a description of modeling of physical, biological and information systems using ordinary differential equations and difference equations Chapter presents a number of examples in some detail, primarily as a reference for problems that will be used throughout the text Following this, Chapter looks at the dynamic behavior of models, including definitions of stability and more complicated nonlinear behavior We provide advanced sections in this chapter on Lyapunov stability analysis because we find that it is useful in a broad array of applications and is frequently a topic that is not introduced until later in one’s studies The remaining three chapters of the first half of the book focus on linear systems, beginning with a description of input/output behavior in Chapter In Chapter 6, we formally introduce feedback systems by demonstrating how state space control laws can be designed This is followed in Chapter by material on output feedback and estimators Chapters and introduce the key concepts of reachability www.elsolucionario.net 387 BIBLIOGRAPHY [Jac95] V Jacobson Congestion avoidance and control ACM SIGCOMM Computer Communication Review, 25:157–173, 1995 [JNP47] H James, N Nichols, and R Phillips Theory of Servomechanisms McGraw-Hill, New York, 1947 [JT61] P D Joseph and J T Tou On linear control theory Transactions of the AIEE, 80(18), 1961 [Jun02] W G Jung, editor Op Amp Applications Analog Devices, Norwood, MA, 2002 [Kal60] R E Kalman Contributions to the theory of optimal control Boletin de la Sociedad Matématica Mexicana, 5:102–119, 1960 [Kal61a] R E Kalman New methods and results in linear prediction and filtering theory Technical Report 61-1, Research Institute for Advanced Studies (RIAS), Baltimore, MD, February 1961 [Kal61b] R E Kalman On 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for automatic controllers Transactions of the ASME, 64:759–768, 1942 www.elsolucionario.net Index access control, see admission control acknowledgment (ack) packet, 78–80 activator, 16, 59, 130 active filter, 155, see also operational amplifier actuators, 4, 31, 51, 66, 82, 179, 225, 267, 286, 314, 328, 337–339, 341 effect on zeros, 286, 338 in computing systems, 76 saturation, 50, 226, 303, 309–311, 314, 328 A/D converters, see analog-to-digital converters adaptation, 300 adaptive control, 21, 377, 378 additive uncertainty, 353, 357, 360, 380 admission control, 54, 64, 79, 80, 276 advertising, 15 aerospace systems, 8–9, 18, 343, see also vectored thrust aircraft; X-29 aircraft AFM, see atomic force microscope aircraft, see flight control alcohol, metabolism of, 95 algebraic loops, 212, 250–251 aliasing, 226 all-pass transfer function, 336 alternating current (AC), 7, 156 amplifier, see operational amplifier amplitude ratio, see gain analog computing, 51, 72, 251, 312 analog implementation, controllers, 75, 264, 312–314 analog-to-digital converters, 4, 83, 225, 226, 314 analytic function, 237 anticipation, in controllers, 6, 24, 300, see also derivative action antiresonance, 157 anti-windup compensation, 310–311, 314, 315, 317 Apache web server, 77, see also web server control apparent volume of distribution, 87, 95 Arbib, M A., 168 argument, of a complex number, 251 arrival rate (queuing systems), 55 artificial intelligence (AI), 12, 20 asymptotes, in Bode plot, 254, 255 asymptotic stability, 42, 103, 104, 106, 107, 113, 115, 118, 119, 121, 141 discrete-time systems, 166 atmospheric dynamics, see environmental science atomic force microscopes, 3, 51, 82–85 contact mode, 82, 157, 200 horizontal positioning, 284, 370 system identification, 258 tapping mode, 82, 293, 302, 307, 332 with preloading, 94 attractor (equilibrium point), 104 www.elsolucionario.net automatic reset, in PID control, 299 automatic tuning, 309, 377 automotive control systems, 6, 22, 51, 70, see also cruise control; vehicle steering autonomous differential equation, 29, see also time-invariant systems autonomous vehicles, 8, 20–21 autopilot, 6, 19, 20 balance systems, 35–37, 49, 171, 189, 242, 338, see also cart-pendulum system; inverted pendulum band-pass filter, 155, 156, 256, 257 bandwidth, 156, 187, 326, 337 Bell Labs, 18, 292 Bennett, S., 25, 292, 316 bicycle dynamics, 70–72, 92, 124, 227 Whipple model, 72 bicycle model, for vehicle steering, 51–53 bicycledynamics Whipple model, 200 bifurcations, 122–125, 131, see also root locus plots biological circuits, 16, 45, 58–60, 130, 167, 257 genetic switch, 64, 115 repressilator, 59–60 biological systems, 1, 3, 10, 16, 22, 25, 58–61, 127, 296, 300, see also biological circuits; drug administration; neural systems; population dynamics bistability, 23, 118 393 INDEX Black, H S., 18, 19, 72, 74, 132, 269, 292, 351 block diagonal systems, 106, 107, 130, 140, 146, 150, 213 block diagram algebra, 243, 246, 360 block diagrams, 1, 44–47, 239, 243–248, 250 control system, 4, 230, 245, 319 Kalman decomposition, 224 observable canonical form, 206 observer, 203, 211 observer-based control system, 214 PID controllers, 296, 299, 314 reachable canonical form, 173 two degree-of-freedom controller, 220, 320, 362 Youla parameterization, 361 Bode, H., 230, 292, 348, 378 Bode plots, 251–258, 285 asymptotic approximation, 254, 255, 265 low-, band-, high-pass filters, 257 nonminimum phase systems, 286 of rational function, 252 sketching, 255 Bode’s ideal loop transfer function, 359, 379 Bode’s integral formula, 339–341, 343–344 Bode’s relations, 285, 330 Brahe, T., 28 breakpoint, 254, 274 Brockett, R W., vii, 1, 164 Bryson, A E., 201 bumpless transfer, 377 Bush, V., 316 calibration, versus feedback, 10, 181, 196, 198 Cannon, R H., 61, 132 capacitor, transfer function for, 237 car, see automotive control systems carrying capacity, in population models, 91 cart-pendulum system, 36, 173, see also balance systems causal reasoning, 1, 71 Cayley-Hamilton theorem, 171, 200, 204 center (equilibrium point), 104 centrifugal governor, 2, 3, 6, 17 chain of integrators (normal form), 61, 174 characteristic polynomial, 105, 200, 236, 241 for closed loop transfer function, 270 observable canonical form, 206 output feedback controller, 213, 214 reachable canonical form, 174, 176, 180, 199 chemical systems, 9, 296, see also process control; compartment models chordal distance, 355 Chrysler autopilot, circuits, see biological circuits; electrical circuits classical control, vi, 378 closed loop, 1, 2, 4, 6, 163, 177, 184, 269, 270, 289, 319 versus open loop, 1, 271, 290, 319 command signals, 4, 22, 221, 296, see also reference signal; setpoint compartment models, 86–90, 108, 152, 187, 204, 209, 228 exercises, 165 compensator, see control law complementary sensitivity function, 321, 329, 341, 354, 358, 360, 364, 369, 373, 378 complexity, of control systems, 9, 21, 301 computed torque, 164 www.elsolucionario.net computer implementation, controllers, 225–227, 314–315 computer science, relationship to control, computer systems, control of, 12–14, 25, 39, 56, 57, 76–81, 158, see also queuing systems conditional integration, 317 conditional stability, 277 congestion control, 12, 78–81, 104, 275, 294, 317, see also queuing systems router dynamics, 94 consensus, 57 control definition of, 3–5 early examples, 2, 5, 6, 8, 10, 18, 22, 25, 299 fundamental limitations, 286, 335–344, 348, 367, 370, 377–378 history of, 25, 316 modeling for, 5, 31–32, 61, 351 successes of, 8, 25 system, 4, 176, 214, 220, 225, 230, 320, 322, 362 using estimated state, 212–215, 374 control error, 23, 245, 297 control law, 4, 23, 24, 163, 177, 180, 245 control Lyapunov function, 125 control matrix, 34, 38 control signal, 31, 158, 296 controllability, 198, see also reachability controlled differential equation, 29, 34, 236 convolution equation, 146–148, 150, 151, 171, 262 discrete-time, 166 coordinate transformations, 107, 148–150, 174, 227, 235–236 to Jordan form, 140 to observable canonical form, 207 394 INDEX to reachable canonical form, 175, 176 Coriolis forces, 36, 164 corner frequency, 254 correlation matrix, 216, 217 cost function, 191 coupled spring-mass system, 143, 145, 149 covariance matrix, 216 critical gain, 306, 308, 309 critical period, 306, 308 critical point, 273, 275, 281, 282, 291, 292, 306, 356, 357, 376 critically damped oscillator, 185 crossover frequency, see gain crossover frequency; phase crossover frequency crossover frequency inequality, see gain crossover frequency inequality cruise control, 6, 17–18, 66–70 Chrysler autopilot, control design, 197, 303, 312 feedback linearization, 163 integrator windup, 309, 310 linearization, 159 pole/zero cancellation, 249 robustness, 18, 351, 352, 358 Curtiss seaplane, 19, 20 cybernetics, 11, see also robotics D/A converters, see digital-to-analog converters damped frequency, 185 damping, 28, 36, 41, 97, 266, 268 damping ratio, 185, 186, 189, 303 DARPA Grand Challenge, 20, 21 DC gain, 156, see also zero frequency gain dead zone, 23 decision making, higher levels of, 8, 12, 19 delay, see time delay delay compensation, 294, 379 delay margin, 283 delta function, see impulse function derivative action, 24, 25, 296, 299–301, 313, 335 filtering, 300, 311–312, 315 setpoint weighting, 312, 315 time constant, 297 versus lead compensator, 334 describing functions, 290–292 design of dynamics, 18–19, 110, 125–127, 132, 168, 178, 183 diabetes, see insulin-glucose dynamics diagonal systems, 106, 140 Kalman decomposition for, 223 transforming to, 107, 130, 139 Dickmanns, E., 20 difference equations, 34, 37–41, 62, 158, 225, 315 differential algebraic equations, 33, see also algebraic loops differential equations, 28, 34–37, 96–99 controlled, 29, 134, 236 equilibrium points, 101–102 existence and uniqueness of solutions, 97–99 first-order, 32, 301 isolated solution, 102 periodic solutions, 102–103, 110 qualitative analysis, 99–103 second-order, 100, 184, 301 solutions, 96, 97, 134, 138, 146, 264 stability, see stability transfer functions for, 237 differential flatness, 222 digital control systems, see computer implementation, controllers digital-to-analog converters, 4, 83, 225, 226, 314 dimension-free variables, 49, 61 direct term, 34, 38, 148, 212, 251 www.elsolucionario.net discrete control, 56 discrete-time systems, 38, 62, 129, 158, 166, 314 Kalman filter for, 216 linear quadratic regulator for, 193 disk drives, 65 disturbance attenuation, 4, 177, 327–328, 362–363 design of controllers for, 323, 324, 330, 341, 349, 373 fundamental limits, 340 in biological systems, 258, 300 integral gain as a measure of, 299, 328, 363 relationship to sensitivity function, 327, 339, 349, 362 disturbance weighting, 376 disturbances, 4, 29, 32, 245, 249, 319, 322, 323 generalized, 375 random, 216 Dodson, B., dominant eigenvalues (poles), 188, 303, 304 double integrator, 138, 169, 237 Doyle, J C., vii, 348, 378 drug administration, 85–90, 95, 152, 187, see also compartment models duality, 208, 212 Dubins car, 53 dynamic compensator, 197, 214 dynamic inversion, 164 dynamical systems, 1, 27, 96, 99, 127 linear, 105, 132 observer as a, 202 state of, 176 stochastic, 216 uncertainty in, 351–353 see also differential equations dynamics matrix, 34, 38, 105, 144 Dyson, F., 27 395 INDEX e-commerce, 13 e-mail server, control of, 39, 158 economic systems, 14–15, 22, 62 ecosystems, 16–17, 90, 182, see also predator-prey system eigenvalue assignment, 177, 179, 181–183, 189, 213, 302, 316 by output feedback, 213 for observer design, 209 eigenvalues, 105, 115, 124, 143, 233 and Jordan form, 140–142, 166 distinct, 129, 130, 139, 145, 223 dominant, 188 effect on dynamic behavior, 184, 186–188, 234 for discrete-time systems, 166 invariance under coordinate transformation, 107 relationship to modes, 143–146 relationship to poles, 240 relationship to stability, 118, 141, 142 eigenvectors, 107, 130, 143, 144 relationship to mode shape, 144 electric power, see power systems (electric) electrical circuits, 33, 45, 75, 132, 237, see also operational amplifier electrical engineering, 6–7, 29–31, 156, 277 elephant, modeling of an, 27 Elowitz, M B., 59 encirclement, 273, see also Nyquist criterion entertainment robots, 12 environmental science, 3, 9, 17 equilibrium points, 91, 101, 105, 133, 160, 169 bifurcations of, 122 discrete time, 62 for closed loop system, 177, 196 for planar systems, 104 region of attraction, 120–122, 129 stability, 103 error feedback, 5, 296, 297, 312, 321 estimators, see oservers381 Euler integration, 41, 42 exponential signals, 231–236, 240, 251 extended Kalman filter, 221 F/A-18 aircraft, Falb, P L., 168 feedback, 1–3 as technology enabler, 2, 19 drawbacks of, 2, 21, 311, 356, 363 in biological systems, 1, 3, 16, 25, 300, see also biological circuits in engineered systems, see control in financial systems, in nature, 3, 15–17, 90 positive, see positive feedback properties, 2, 5, 17–23, 319, 324, 351 robustness through, 17 versus feedforward, 22, 299, 324 feedback connection, 244, 289, 290 feedback controller, 245, 319 feedback linearization, 162–164 feedback loop, 4, 269, 319, 362 feedback uncertainty, 353, 360 feedforward, 22, 220–223, 245, 319, 323, 325 Fermi, E., 27 filters active, 155 for disturbance weighting, 377 for measurement signals, 21, 226, 363 see also band-pass filters; high-filters; low-pass www.elsolucionario.net filters financial systems, see economic systems finite escape time, 98 finite state machine, 70, 77 first-order systems, 135, 166, 237, 253, 254 fisheries management, 95 flatness, see differential flatness flight control, 8, 18, 19, 53, 164 airspace management, F/A-18 aircraft, X-29 aircraft, 341 X-45 aircraft, see also vectored thrust aircraft flow, of a vector field, 29, 100 flow in a tank, 128 flow model (queuing systems), 55, 294, 317 flyball governor, see centrifugal governor force feedback, 10, 11 forced response, 134, 232 Forrester, J W., 15 Fourier, J B J., 61, 263 frequency domain, 230–232, 269, 287, 319 frequency response, 30, 43, 44, 153–158, 231, 292, 306, 326 relationship to Bode plot, 251 relationship to Nyquist plot, 272, 274 second-order systems, 186, 257 system identification using, 258 fully actuated systems, 241 fundamental limits, see control: fundamental limitations Furuta pendulum, 131 gain, 24, 43, 74, 154, 155, 187, 231, 235, 240, 251, 281, 287–291, 351 H∞ , 289, 375 observer, see observer gain 396 INDEX of a system, 288 reference, 196 state feedback, 177, 178, 181, 196, 198 zero frequency, see zero frequency gain see also integral gain gain crossover frequency, 281, 282, 326, 331, 336, 355, 369 gain crossover frequency inequality, 336, 338 gain curve (Bode plot), 251–255, 285, 331 gain margin, 281–283 from Bode plot, 282 reasonable values, 283 gain scheduling, 221, 377 gain-bandwidth product, 75, 238, 365 Gang of Four, 321, 348, 362 Gang of Six, 321, 326 gene regulation, 16, 58, 59, 167, 257 genetic switch, 64, 115, 116 global behavior, 104, 121–125 Glover, K., 348, 378 glucose regulation, see insulin-glucose dynamics Golomb, S., 66 governor, see centrifugal governor H∞ control, 375–378, 380 Harrier AV-8B aircraft, 53 heat propagation, 239 Heaviside, O., 164 Heaviside step function, 151, 164 Hellerstein, J L., 13, 25, 81 high-frequency roll-off, 330, 363, 370 high-pass filter, 256, 257 Hill function, 59 Hoagland, M B., Hodgkin-Huxley equations, 60 homeostasis, 3, 58 homogeneous solution, 134, 137, 138, 240 Honeywell thermostat, Horowitz, I M., 227, 348, 373, 378 human-machine interface, 66, 70 hysteresis, 23, 291, 292 identification, see system identification impedance, 237, 313 implementation, controllers, see analog implementation; computer implementation impulse function, 147, 165, 170 impulse response, 136, 147, 148, 262 inductor, transfer function for, 237 inertia matrix, 36, 164 infinity norm, 289, 376 information systems, 12, 54–58, see also congestion control; web server control initial condition, 97, 100, 103, 133, 138, 145, 216 initial condition response, 134, 137–140, 144, 145, 148, 232 initial value problem, 97 inner loop control, 345, 347 input sensitivity function, see load sensitivity function input/output models, 5, 29–31, 133, 146–159, 230, 288, see also frequency response; steady-state response; step response and transfer functions, 262 and uncertainty, 51, 353 from experiments, 258 relationship to state space models, 32, 96, 147 steady-state response, 150 transfer function for, 236 inputs, 29, 32 insect flight control, 46–47 instrumentation, 10–11, 72 insulin-glucose dynamics, 1, 89–90 integral action, 24–26, 196–199, 296, 298–299, 301, 328 www.elsolucionario.net for bias compensation, 227 setpoint weighting, 312, 315 time constant, 297 integral gain, 24, 297, 299, 302 integrator windup, 226, 309–311, 317 conditional integration, 317 intelligent machines, see robotics internal model principle, 215, 222 Internet, 12, 13, 76, 78, 81, 94, see also congestion control Internet Protocol (IP), 78 invariant set, 119, 122 inverse model, 163, 220, 324 inverse response, 286, 294 inverted pendulum, 37, 71, 101, 109, 119, 122, 129, 131, 278, 341, see also balance systems Jacobian linearization, 160–162 Jordan form, 140–143, 166, 189 Kalman, R E., 168, 198, 202, 224, 227 Kalman decomposition, 223–225, 236, 263, 265 Kalman filter, 216–219, 227, 374 extended, 221 Kalman-Bucy filter, 218 Kelly, F P., 81 Kepler, J., 28 Keynes, J M., 14 Keynesian economic model, 62, 167 Krasovski-Lasalle principle, 119 LabVIEW, 124, 165 lag, see phase lag lag compensation, 330, 332 Laplace transforms, vi, 260–263 Laplacian matrix, 58 Lasalle’s invariance principle, see Krasovski-Lasalle principle 397 INDEX lead, see phase lead lead compensation, 332, 334, 335, 345, 350 limit cycle, 92, 102, 103, 110, 112, 123, 124, 290, 291 linear quadratic control, 191–195, 217, 227, 373–375 linear systems, 30, 34, 75, 105, 132–165, 223, 232, 236, 263, 288 linear time-invariant systems, 30, 34, 135, 262 linearity, 134, 251 linearization, 110, 118, 133, 159–164, 221, 351 Lipschitz continuity, 99 load disturbances, 319, 363, see also disturbances load sensitivity function, 321 local behavior, 104, 110, 119, 121, 160 locally asymptotically stable, 104 logistic growth model, 90, 91, 95 loop analysis, 269, 319 loop shaping, 272, 330–335, 347, 373 design rules, 331 fundamental limitations, 335–344 see also Bode’s loop transfer function loop transfer function, 269–272, 281, 282, 289, 319, 322, 330, 334, 340, 348, see also Bode’s loop transfer function Lotus Notes server, see e-mail server low-order models, 301 low-pass filter, 256, 257, 311 LQ control, see linear quadratic control LTI systems, see linear time-invariant systems Lyapunov equation, 115, 129 Lyapunov functions, 112–115, 122, 128, 165 design of controllers using, 119, 125 existence of, 114 Lyapunov stability analysis, 43, 111–121, 127 discrete time, 129 manifold, 121 margins, see stability margins Mars Exploratory Rovers, 11, 12 mass spectrometer, 10 materials science, Mathematica, 41, 124, 165 MATLAB, 26, 41, 124, 165, 201 acker, 182, 212 dlqe, 217 dlqr, 195 hinfsyn, 376 jordan, 140 linmod, 161 lqr, 192 place, 182, 190, 212 trim, 161 matrix exponential, 137–140, 144, 146, 164, 165 coordinate transformations, 149 Jordan form, 141 second-order systems, 139, 165 maximum complementary sensitivity, 358, 369 maximum sensitivity, 327, 356, 370 measured signals, 31, 32, 34, 96, 202, 214, 226, 320, 322, 375 measurement noise, 4, 21, 202, 204, 216, 218, 245, 311, 319–321, 330, 363 response to, 328–330, 363 mechanical systems, 31, 35, 42, 51, 61, 164 mechanics, 28–29, 31, 127, 132 minimal model (insulin-glucose), 89, 90, see also insulin-glucose dynamics minimum phase, 285, 292, 335 modal form, 131, 146, 150 Modelica, 33 www.elsolucionario.net modeling, 5, 27–33, 61, 66 control perspective, 31 discrete control, 56 discrete-time, 37–38, 158–159 frequency domain, 230–232 from experiments, 47–48 model reduction, normalization and scaling, 49 of uncertainty, 50–51 simplified models, use of, 32, 301, 352, 358, 359 software for, 33, 161, 164 state space, 34–43 uncertainty, see uncertainty modes, 143–145, 240 relationship to poles, 242 motion control systems, 51–54, 227 motors, electric, 64, 200, 228, 229 multi-input, multi-output systems, 289, 322, 331, see also input/output models multiplicative uncertainty, 353, 360 nanopositioner (AFM), 284, 370 natural frequency, 185, 303 negative definite function, 112 negative feedback, 18, 22, 74, 177, 269, 300 Nernst’s law, 61 networking, 12, 45, 81, see also congestion control neural systems, 10, 47, 60, 300 neutral stability, 103, 104 Newton, I., 28 Nichols, N B., 164, 305, 347 Nichols chart, 373, 374 Nobel Prize, 10, 11, 14, 61, 82 noise, see disturbances; measurement noise noise attenuation, 258, 328–330 noise cancellation, 125 noise sensitivity function, 321 nonlinear systems, 31, 96, 99, 102, 109, 111, 115, 398 INDEX 121–127, 203, 221, 288–290 linear approximation, 110, 118, 160, 166, 351 system identification, 63 nonminimum phase, 285, 286, 294, 335–337, see also inverse response nonunique solutions (ODEs), 98 normalized coordinates, 49–50, 63, 162 norms, 288–289 Nyquist, H., 269, 292 Nyquist criterion, 273, 275, 278, 280, 289, 290, 306 for robust stability, 356, 380 Nyquist D contour, 272, 278 Nyquist plot, 272–273, 281, 306, 328, 374 optimal control, 191, 216, 218, 374 order, of a system, 34, 236 ordinary differential equations, see differential equations oscillator dynamics, 93, 97, 98, 139, 185, 234, 237 normal form, 63 see also nanopositioner (AFM); spring-mass system outer loop control, 345–347 output feedback, 212, 213, 227, see also control: using estimated state; loop shaping; PID control output sensitivity function, see noise sensitivity function outputs, see measured signals overdamped oscillator, 185 overshoot, 152, 177, 186, 326 observability, 32, 202–203, 223, 227 rank condition, 204 tests for, 203–204 unobservable systems, 205, 223–224, 266 observability matrix, 204, 206 observable canonical form, 205, 206, 227 observer gain, 208, 210–212, 214, 216–218 observers, 202, 207–210, 218, 221 block diagram, 203, 211 see also Kalman filter ODEs, see differential equations Ohm’s law, 60, 74, 237 on-off control, 23, 24 open loop, 1, 2, 74, 169, 246, 269, 309, 319, 327, 353 open loop gain, 238, 281, 326 operational amplifiers, 72–76, 238, 312, 360 circuits, 93, 155, 270, 364 dynamic model, 75, 238 input/output characteristics, 73 oscillator using, 93, 130 static model, 73, 238 Padé approximation, 294, 337 paging control (computing), 56 parallel connection, 244 parametric stability diagram, 123, 125 parametric uncertainty, 51, 351 particle accelerator, 11 particular solution, 134, 153, see also forced response passive systems, 290, 340 passivity theorem, 290 patch clamp, 10 PD control, 299, 332 peak frequency, 157, 326 pendulum dynamics, 114, see also inverted pendulum perfect adaptation, 300 performance, 77 performance limitations, 335, 340, 369, 377 due to right half-plane poles and zeros, 286 see also control: fundamental limitations performance specifications, 152, 176, 319, 326–331, 362, see also overshoot; maximum sensitivity; resonant peak; rise time; settling time www.elsolucionario.net periodic solutions, see differential equations; limit cycles persistence, of a web connection, 77, 78 Petri net, 45 pharmacokinetics, 86, 90, see also drug administration phase, 43, 154, 155, 187, 231, 235, 251, 291, see also minimum phase; nonminimum phase minimum vs nonminimum, 285 phase crossover frequency, 281, 282 phase curve (Bode plot), 251–253, 255 relationship to gain curve, 285, 330 phase lag, 154, 155, 257, 285, 336, 337 phase lead, 154, 257, 335, 350 phase margin, 281, 282, 331, 333, 336, 350, 379 from Bode plot, 282 reasonable values, 283 phase portrait, 28, 29, 99–101, 121 Philbrick, G A., 76 photoreceptors, 300 physics, relationship to control, PI control, 17, 25, 66, 69, 299, 304, 332 first-order system, 302, 368 PID control, 24–25, 236, 296–316, 335 block diagram, 297, 299, 311 computer implementation, 314 ideal form, 296, 316 implementation, 299, 311–315 in biological systems, 300 op amp implementation, 312–314 tuning, 305–309 see also derivative action; integral action pitchfork bifurcation, 131 399 INDEX planar dynamical systems, 100, 104, see also second-order systems pole placement, 177, 365, 369–370, see also eigenvalue assignment robust, 365 pole zero diagram, 241 pole/zero cancellations, 248–250, 266, 369, 370 poles, 240, 242 dominant, 304, see also dominant eigenvalues (poles) fast stable, 368, 370 pure imaginary, 272, 278 relationship to eigenvalues, 240 right half-plane, 242, 278, 286, 335, 337–338, 340, 350, 370 population dynamics, 90–92, 95, see also predator-prey system positive definite function, 112, 113, 115, 119 positive definite matrix, 115, 192 positive feedback, 16, 21–23, 130, 299 positive real (transfer function), 340 power of a matrix, 137 power systems (electric), 6–7, 63, 102, 128 predator-prey system, 38, 91–92, 123, 182 prediction, in controllers, 24, 25, 221, 299, 379, see also derivative action prediction time, 300 principle of the argument, see variation of the argument, principle of process control, 9, 10, 13, 45 proportional control, 24, 296, see also PID control proportional, integral, derivative control, see PID control protocol, see congestion control; consensus pulse signal, 147, 148, 188, see also impulse function pupil response, 259, 300 pure exponential response, 233 Q-value, 63, 187, 255 quantitative feedback theory (QFT), 373 quarter car model, 266, 267 queuing systems, 54–56, 64 random process, 55, 216, 229 reachability, 32, 168–176, 198, 223 rank condition, 171 tests for, 170 unreachable systems, 172, 200, 223–224, 266 reachability matrix, 170, 174 reachable canonical form, 35, 173–176, 179, 181, 199 reachable set, 168 real-time systems, reference signal, 23, 176, 177, 230, 245, 296, 312, 321, 323, see also command signals; setpoint effect on observer error, 213, 220, 225 response to, 326, 349 tracking, 176, 220, 221, 330, 364 reference weighting, see setpoint weighting region of attraction, see equilibrium points: regions of attraction regulator, see control law relay feedback, 291, 308 Reno (protocol), see Internet; congestion control repressilator, 59–60 repressor, 16, 59, 64, 115, 167, 258 reset, in PID control, 298, 299 resonant frequency, 187, 289 resonant peak, 157, 187, 326, 359 resource usage, in computing systems, 13, 55, 57, 76, 77 response, see input/output models www.elsolucionario.net retina, 300, see also pupil response Riccati equation, 192, 218, 376, 378 Riemann sphere, 355 right half-plane poles and zeros, see poles: right half-plane; zeros: right half-plane rise time, 152, 177, 186, 326 robotics, 8, 11–12, 164 robustness, 17–18, 326, 353, 378 performance, 362–365, 373–378 stability, 356–362 using gain and phase margin, 283, 330 using maximum sensitivity, 327, 330, 357, 379, 380 using pole placement, 365–372 via gain and phase margin, 282 see also uncertainty roll-off, see high-frequency roll-off root locus diagram, 124 Routh-Hurwitz criterion, 131 rush-hour effect, 56, 64 saddle (equilibrium point), 104 sampling, 158, 225, 226, 314 saturation function, 45, 73, 314, see also actuators: saturation scaling, see normalized coordinates scanning tunneling microscope, 11, 82 schematic diagrams, 44, 45, 72 Schitter, G., 84, 85 second-order systems, 28, 165, 184–188, 201, 254, 304 Segway Personal Transporter, 35, 171 self-activation, 130 self-repression, 167, 257 semidefinite function, 112 sensitivity crossover frequency, 328 400 INDEX sensitivity function, 321, 328, 329, 331, 340, 357, 364, 370 and disturbance attenuation, 327, 340, 349 sensor matrix, 34, 38 sensor networks, 57 sensors, 4, 9, 203, 225, 286, 314, 319, 322, 337, 338, 375 effect on zeros, 286, 338 in computing systems, 76 see also measured signals separation principle, 202, 214 series connection, 243, 244 service rate (queuing systems), 55 setpoint, 296 setpoint weighting, 312, 315 settling time, 152, 166, 177, 186, 326 similarity of two systems, 353–356 simulation, 40–42, 51 SIMULINK, 161 single-input, single-output (SISO) systems, 96, 133, 134, 160, 205, 288 singular values, 288, 289, 380 sink (equilibrium point), 104 small gain theorem, 289–290, 359 Smith predictor, 379 software tools for control, v solution (ODE), see differential equations: solutions Sony AIBO, 12 source (equilibrium point), 104 spectrum analyzer, 258 Sperry autopilot, 19 spring-mass system, 28, 40, 42, 43, 83, 128 coupled, 145, 149 generalized, 35, 72 identification, 47 normalization, 49, 63 see also oscillator dynamics stability, 4, 5, 18, 19, 42, 99, 103–121 asymptotic stability, 103, 107 conditional, 277 in the sense of Lyapunov, 103 local versus global, 104, 111, 121, 122 Lyapunov analysis, see Lyapunov stability analysis neutrally stable, 103, 104 of a system, 106 of equilibrium points, 42, 103, 104, 112, 113, 118 of feedback loop, see Nyquist criterion of limit cycles, 110 of linear systems, 105–109, 115, 141 of solutions, 103, 111 of transfer functions, 241 robust, see robust stability unstable solutions, 104 using eigenvalues, 118, 141, 142 using linear approximation, 109, 118, 161 using Routh-Hurwitz criterion, 131 using state feedback, 176–195 see also bifurcations; equilibrium points stability diagram, see parametric stability diagram stability margin (quantity), 282, 283, 327, 350, 357, 376 reasonable values, 283 stability margins (concept), 280–285, 294, 330 stable pole, 242 stable zero, 242 Stark, L., 259 state, of a dynamical system, 28, 31, 34 state estimators, see observers state feedback, 168–198, 208, 213, 220–222, 225–227, 366, 374, see also eigenvalue assignment; linear quadratic control state space, 28, 34–43, 176 www.elsolucionario.net state vector, 34 steady-state gain, see zero frequency gain steady-state response, 26, 42, 150–158, 166, 177, 186, 231, 232, 234, 258, 263 steam engines, 2, 3, 17 steering, see vehicle steering Stein, G., vii, 1, 319, 341 step input, 30, 136, 151, 240, 305 step response, 30, 31, 47, 48, 136, 148, 151, 152, 177, 185, 186, 305 stochastic cooling, 11 stochastic systems, 216, 218 summing junction, 45 superposition, 30, 134, 148, 165, 231 supervisory control, see decision making: higher levels of supply chains, 15 supremum (sup), 288 switching behavior, 22, 64, 118, 377 system identification, 47, 62, 258 tapping mode, see atomic force microscope TCP/IP, see Internet; congestion control Teorell, T., 86, 90 thermostat, 5, three-term controllers, 296, see also PID control thrust vectored aircraft, see vectored thrust aircraft time constant, first-order system, 166 time delay, 6, 13, 236, 237, 283, 285, 306, 314, 337, 338 compensation for, 379 Padé approximation, 294, 337 time plot, 28 time-invariant systems, 30, 34, 127, 135–136 tracking, see reference signal: tracking 401 INDEX trail (bicycle dynamics), 71, 72 transcriptional regulation, see gene regulation transfer functions, 230–263 by inspection, 236 derivation using exponential signals, 232 derivation using Laplace transforms, 262 for control systems, 245, 265 for electrical circuits, 237 for time delay, 236 frequency response, 231, 251 from experiments, 258 irrational, 237, 240 linear input/output systems, 232, 236, 265 transient response, 42, 150, 152, 154, 169, 189, 232, 233 Transmission Control Protocol (TCP), 78 transportation systems, Tsien, H S., 11 tuning rules, 317, see Ziegler-Nichols tuning two degree-of-freedom control, 220, 297, 323, 325, 348, 349 uncertainty, 4, 17–18, 32, 50–51, 196, 351–356 component or parameter variation, 4, 50, 351 disturbances and noise, 4, 32, 176, 245, 319 unmodeled dynamics, 4, 50, 352, 357 see also additive uncertainty; feedback uncertainty; multiplicative uncertainty uncertainty band, 50 uncertainty lemon, 50, 51, 69, 75, 85 underdamped oscillator, 98, 185, 186 unit step, 151 unmodeled dynamics, see uncertainty: unmodeled dynamics unstable pole, see poles: right half-plane unstable pole/zero cancellation, 249 unstable solution, for a dynamical system, 104, 107, 142, 242 unstable zero, see zeros: right half-plane variation of the argument, principle of, 279, 292 vector field, 29, 100 vectored thrust aircraft, 53–54, 142, 192, 218, 265, 334, 344 vehicle steering, 51–53, 161, 178, 210, 215, 222, 246, 286, 293, 325, 366 ship dynamics, 52 vehicle suspension, 266, see also coupled spring-mass system vertical takeoff and landing, see vectored thrust aircraft vibration absorber, 267 Vinnicombe, G., 348, 355, 356, 378 Vinnicombe metric, 353–356, 376 voltage clamp, 10, 11, 61 www.elsolucionario.net waterbed effect, 340, 341 Watt governor, see centrifugal governor Watt steam engine, 2, 17 web server control, 76–78, 193 web site, companion, v Whipple, F J W., 72 Wiener, N., 11, 12 winding number, 279 window size (TCP), 79, 81, 105 windup, see integrator windup Wright, W., 18 Wright Flyer, 8, 19 X-29 aircraft, 341 X-45 aircraft, Youla parameterization, 360–362 zero frequency gain, 156, 178, 181, 187, 240 zeros, 240 Bode plot for, 265 effect of sensors and actuators on, 286, 287, 338 for a state space system, 241 right half-plane, 242, 286, 335–338, 341, 350, 369 signal-blocking property, 240 slow stable, 366, 367, 369 Ziegler, J G., 305, 316 Ziegler-Nichols tuning, 305–308, 316 frequency response, 306 improved method, 306 step response, 305 ...www.elsolucionario.net Feedback Systems An Introduction for Scientists and Engineers ˚ Karl Johan Aström Richard M Murray Version v2.10e (30 August 2011) This is the electronic edition of Feedback Systems and... dramatically improved performance Force feedback is also used in haptic devices for manual control Another important application of feedback is in instrumentation for biological systems Feedback is widely... http://www.cds.caltech.edu/ ^murray/ FBSwiki www.elsolucionario.net Contents Preface vi Chapter Introduction 1.1 What Is Feedback? 1.2 What Is Control? 1.3 Feedback Examples 1.4 Feedback Properties 1.5 Simple Forms of Feedback